The analytic–synthetic distinction is a semantic distinction used primarily in philosophy to distinguish between propositions (in particular, statements that are affirmative subject–predicate judgments) that are of two types: analytic propositions and synthetic propositions. Analytic propositions are true or not true solely by virtue of their meaning, whereas synthetic propositions' truth, if any, derives from how their meaning relates to the world.
The distinction between analytic and synthetic method in various areas of knowledge, in particular in mathematics, dates to antiquity. However, the distinction between analytic and synthetic propositions was first proposed by Immanuel Kant. It was then revised considerably over time, and different philosophers have used the terms in very different ways. Furthermore, some philosophers (starting with Willard Van Orman Quine) have questioned whether there is even a clear distinction to be made between propositions which are analytically true and propositions which are synthetically true. Debates regarding the nature and usefulness of the distinction continue to this day in contemporary philosophy of language.
Kant
Conceptual containment The philosopher Immanuel Kant uses the terms "analytic" and "synthetic" to divide propositions into two types. Kant introduces the analytic–synthetic distinction in the Introduction to his Critique of Pure Reason (1781/1998, A6–7/B10–11). There, he restricts his attention to statements that are affirmative subject–predicate judgments and defines "analytic proposition" and "synthetic proposition" as follows: analytic proposition: a proposition whose predicate concept is contained in its subject concept synthetic proposition: a proposition whose predicate concept is not contained in its subject concept but related
Examples of analytic propositions, on Kant's definition, include: "All bachelors are unmarried." "All triangles have three sides."
Kant's own example is: "All bodies are extended": that is, they occupy space. (A7/B11)
Each of these statements is an affirmative subject–predicate judgment, and, in each, the predicate concept is contained within the subject concept. The concept "bachelor" contains the concept "unmarried"; the concept "unmarried" is part of the definition of the concept "bachelor". Likewise, for "triangle" and "has three sides", and so on.
Examples of synthetic propositions, on Kant's definition, include: "All bachelors are alone." "All creatures with hearts have kidneys."
Kant's own example is: "All bodies are heavy": that is, they experience a gravitational force. (A7/B11)
As with the previous examples classified as analytic propositions, each of these new statements is an affirmative subject–predicate judgment. However, in none of these cases does the subject concept contain the predicate concept. The concept "bachelor" does not contain the concept "alone"; "alone" is not a part of the definition of "bachelor". The same is true for "creatures with hearts" and "have kidneys"; even if every creature with a heart also has kidneys, the concept "creature with a heart" does not contain the concept "has kidneys". So the philosophical issue is: What kind of statement is "Language is used to transmit meaning"?
Kant's version and the a priori–a posteriori distinction
In the Introduction to the Critique of Pure Reason, Kant contrasts his distinction between analytic and synthetic propositions with another distinction, the distinction between a priori and a posteriori propositions. He defines these terms as follows: a priori proposition: a proposition whose justification does not rely upon experience. Moreover, the proposition can be validated by experience, but is not grounded in experience. Therefore, it is logically necessary. a posteriori proposition: a proposition whose justification does rely upon experience. The proposition is validated by, and grounded in, experience. Therefore, it is logically contingent.
Examples of a priori propositions include: "All bachelors are unmarried." "7 + 5 = 12."
The justification of these propositions does not depend upon experience: one need not consult experience to determine whether all bachelors are unmarried, nor whether . (Of course, as Kant would grant, experience is required to understand the concepts "bachelor", "unmarried", "7", "+" and so forth. However, the a priori–a posteriori distinction as employed here by Kant refers not to the origins of the concepts but to the justification of the propositions. Once we have the concepts, experience is no longer necessary.)
Examples of a posteriori propositions include: "All bachelors are unhappy." "Tables exist."
Both of these propositions are a posteriori: any justification of them would require one's experience.
The analytic–synthetic distinction and the a priori–a posteriori distinction together yield four types of propositions: analytic a priori synthetic a priori analytic a posteriori synthetic a posteriori
Kant posits the third type as obviously self-contradictory. Ruling it out, he discusses only the remaining three types as components of his epistemological frameworkeach, for brevity's sake, becoming, respectively, "analytic", "synthetic a priori", and "empirical" or "a posteriori" propositions. This triad accounts for all propositions possible. Examples of analytic and examples of a posteriori statements have already been given, for synthetic a priori propositions he gives those in mathematics and physics.
The ease of knowing analytic propositions Part of Kant's argument in the Introduction to the Critique of Pure Reason involves arguing that there is no problem figuring out how knowledge of analytic propositions is possible. To know an analytic proposition, Kant argued, one need not consult experience. Instead, one needs merely to take the subject and "extract from it, in accordance with the principle of contradiction, the required predicate" (B12). In analytic propositions, the predicate concept is contained in the subject concept. Thus, to know an analytic proposition is true, one need merely examine the concept of the subject. If one finds the predicate contained in the subject, the judgment is true.
Thus, for example, one need not consult experience to determine whether "All bachelors are unmarried" is true. One need merely examine the subject concept ("bachelors") and see if the predicate concept "unmarried" is contained in it. And in fact, it is: "unmarried" is part of the definition of "bachelor" and so is contained within it. Thus the proposition "All bachelors are unmarried" can be known to be true without consulting experience.
It follows from this, Kant argued, first: All analytic propositions are a priori; there are no a posteriori analytic propositions. It follows, second: There is no problem understanding how we can know analytic propositions; we can know them because we only need to consult our concepts in order to determine that they are true.
The possibility of metaphysics After ruling out the possibility of analytic a posteriori propositions, and explaining how we can obtain knowledge of analytic a priori propositions, Kant also explains how we can obtain knowledge of synthetic a posteriori propositions. That leaves only the question of how knowledge of synthetic a priori propositions is possible. This question is exceedingly important, Kant maintains, because all scientific knowledge (for him Newtonian physics and mathematics) is made up of synthetic a priori propositions. If it is impossible to determine which synthetic a priori propositions are true, he argues, then metaphysics as a discipline is impossible. The remainder of the Critique of Pure Reason is devoted to examining whether and how knowledge of synthetic a priori propositions is possible.See Cooper Harold Langford (1949)'s ostensive proof:
Mathematics and Synthetic Apriori Propositions.
One example Kant gives of a possibly synthetic a priori propositions are the propositions of mathematics. The mathematical equation that 10 = 0.2x 50 is true regardless of experience thus making it a priori, but not analytic. Mathematical propositions are not analytic in that 10 does not self evidently contain 0.2x50, in the same way that the concept bachelor contains the categories of unmarried and male.
The Importance of Synthetic Apriori Propositions to Kant's metaphysics
Kant's advocacy for his metaphysics in Critique of Pure Reason can be seen as relying on the possibility of synthetic apriori claims. If synthetic apriori propositions are possible, it supposes a certain metaphysical worldview, much of the Critique of Pure reason then relies on the possibility of synthetic apriori propositions to justify a worldview. One could reduce Kant's argument into a simple form: If Kant's metaphysics is true, then synthetic apriori propositions are possible.
Frege and the logical positivists
Frege revision of Kantian definition Over a hundred years later, a group of philosophers took interest in Kant and his distinction between analytic and synthetic propositions: the logical positivists.
Part of Kant's examination of the possibility of synthetic a priori knowledge involved the examination of mathematical propositions, such as "7 + 5 = 12." (B15–16) "The shortest distance between two points is a straight line." (B16–17)
Kant maintained that mathematical propositions such as these are synthetic a priori propositions, and that we know them. That they are synthetic, he thought, is obvious: the concept "equal to 12" is not contained within the concept "7 + 5"; and the concept "straight line" is not contained within the concept "the shortest distance between two points". From this, Kant concluded that we have knowledge of synthetic a priori propositions.
Although not strictly speaking a logical positivist, Gottlob Frege's notion of analyticity influenced them greatly. It included a number of logical properties and relations beyond containment: symmetry, transitivity, antonymy, or negation and so on. He had a strong emphasis on formality, in particular formal definition, and also emphasized the idea of substitution of synonymous terms. "All bachelors are unmarried" can be expanded out with the formal definition of bachelor as "unmarried man" to form "All unmarried men are unmarried", which is recognizable as tautologous and therefore analytic from its logical form: any statement of the form "All X that are (F and G) are F". Using this particular expanded idea of analyticity, Frege concluded that Kant's examples of arithmetical truths are analytical a priori truths and not synthetic a priori truths.
(Here "logical empiricist" is a synonym for "logical positivist".)
The origin of the logical positivist's distinction The logical positivists agreed with Kant that we have knowledge of mathematical truths, and further that mathematical propositions are a priori. However, they did not believe that any complex metaphysics, such as the type Kant supplied, are necessary to explain our knowledge of mathematical truths. Instead, the logical positivists maintained that our knowledge of judgments like "all bachelors are unmarried" and our knowledge of mathematics (and logic) are in the basic sense the same: all proceeded from our knowledge of the meanings of terms or the conventions of language.
Logical positivist definitions Thus the logical positivists drew a new distinction, and, inheriting the terms from Kant, named it the "analytic-synthetic distinction". They provided many different definitions, such as the following: analytic proposition: a proposition whose truth depends solely on the meaning of its terms analytic proposition: a proposition that is true (or false) by definition analytic proposition: a proposition that is made true (or false) solely by the conventions of language